Uitwerkingen! Hoofdstuk 17 Basisboek wiskunde Craats en Bosch

Uitwerkingen Hoofdstuk 17

Wiskunde 2



17.1

a. Je weet: 2𝜋 = 360° → maak gebruik van deze verhouding!
1 1
Gevraagd: hoeveel rad = 30° → 30° = ∗ 360° → 𝑑𝑎𝑛 𝑖𝑠 𝑏𝑖𝑗𝑏𝑒ℎ𝑜𝑟𝑒𝑛𝑑𝑒 𝑎𝑎𝑛𝑡. 𝑟𝑎𝑑: ∗ 2𝜋
12 12

1 𝜋
→ 12 ∗ 2𝜋 = 6 Oftewel, als kruistabel: 360° 30°
360
360° 30° 2𝜋 360 1 𝟐𝝅∗𝟑𝟎 ? rad
→ = → = 2𝜋∗30 = ? 𝑟𝑎𝑑 → =? 𝒓𝒂𝒅 2π rad gevraagde hoek
2𝜋 ? 30 𝟑𝟔𝟎
in rad
2𝜋∗15 2𝜋∗5 2𝜋 𝜋
Vereenvoudigen geeft: = = 12 =
180 60 6

b. Maak steeds gebruik van dezelfde methode, maar dan alleen de dikgedrukte vergelijking uit a
2𝜋∗45° 2𝜋∗15 2𝜋∗5 2𝜋 𝜋
=? 𝑟𝑎𝑑 = = = =
360 120 40 8 4

2𝜋∗60° 2𝜋∗6 2𝜋 𝜋
c. =? 𝑟𝑎𝑑 = = =3
360 36 6

dit is 2x 30° (uit a.) en dus ook 2x zoveel radialen (evenredige verhouding (lineair))
𝜋∗70°
2𝜋∗70° 𝜋∗70° 𝜋∗7
d. =? 𝑟𝑎𝑑 = = 𝑑𝑒𝑙𝑒𝑛 𝑑𝑜𝑜𝑟 10 = 10
180° =
360° 180° 18
10
2𝜋∗10 𝜋 7𝜋
𝑂𝑝 𝑎𝑛𝑑𝑒𝑟𝑒 𝑚𝑎𝑛𝑖𝑒𝑟: 𝑏𝑒𝑝𝑎𝑎𝑙 𝑒𝑒𝑟𝑠𝑡 10° → = 18 → dan x7 = 18 = zelfde
360

2𝜋∗15° 2𝜋∗5° 10𝜋 𝜋
e. =? 𝑟𝑎𝑑 = = = 12
360 120 120

Ik raad je aan om het aantal radialen van 30°, 45° en 90° uit je hoofd te leren.



17.2

2𝜋∗20° 40𝜋 20𝜋 20𝜋 10𝜋 𝜋
a. 360
=? 𝑟𝑎𝑑 = 360 = 180
= 180 = 90
= 9

2𝜋∗50° 100𝜋 10𝜋 5𝜋
b. =? 𝑟𝑎𝑑 = = = 18
360 360 36

2𝜋∗80° 160𝜋 80𝜋 40𝜋 20𝜋 4𝜋
c. =? 𝑟𝑎𝑑 = = = = =
360 360 180 90 45 9

2𝜋∗100° 200𝜋 20𝜋 10𝜋 5𝜋
d. =? 𝑟𝑎𝑑 = = = =
360 360 36 18 9

2𝜋∗150° 300𝜋 100𝜋 10𝜋 5𝜋
e. =? 𝑟𝑎𝑑 = = = =
360 360 120 12 6

,17.3

2𝜋∗130° 260𝜋 26𝜋 13𝜋
a. =? 𝑟𝑎𝑑 = = =
360 360 36 18

2𝜋∗135° 270𝜋 27𝜋 3𝜋
b. =? 𝑟𝑎𝑑 = = =
360 360 36 4

2𝜋∗200° 400𝜋 40𝜋 10𝜋
c. =? 𝑟𝑎𝑑 = = =
360 360 36 9

2𝜋∗240° 480𝜋 48𝜋 24𝜋 12𝜋 4𝜋
d. =? 𝑟𝑎𝑑 = = = = =
360 360 36 18 9 3

2𝜋∗330° 660𝜋 66𝜋 22𝜋 11𝜋
e. =? 𝑟𝑎𝑑 = = = =
360 360 36 12 6




17.4

a.
360° ?°

2π rad 1/6 π
gevraagde hoek in rad


360° ? 360 𝜋 360𝜋 180
= 𝜋 → ∗6 =? = = = 30°
2𝜋 2𝜋 12𝜋 6
6

360° ? 360 7𝜋 7∗360𝜋 7∗180
b. =7 → ∗ 6 =? = = = 7 ∗ 30 = 210°
2𝜋 𝜋 2𝜋 12𝜋 6
6

360° ? 360 1𝜋 360𝜋 360
c. =1 → ∗ =? = = = 60°
2𝜋 𝜋 2𝜋 3 6𝜋 6
3

360° ? 360 2𝜋 2∗360𝜋 360
d. =2 → ∗ 3 =? = = = 120°
2𝜋 𝜋 2𝜋 6𝜋 3
3

360° ? 360 1𝜋 360𝜋 360 180 90
e. =1 → ∗ 4 =? = = = = = 45°
2𝜋 𝜋 2𝜋 8𝜋 8 4 2
4




17.5
360° ? 360 5𝜋 5∗360𝜋 5∗180
a. =5 → ∗ 4 =? = = = 5 ∗ 45 = 225°
2𝜋 𝜋 2𝜋 8𝜋 4
4

360° ? 360 5𝜋 5∗360𝜋 5∗180 5∗90
b. = 5 → ∗ 12 =? = = = = 5 ∗ 15 = 75°
2𝜋 𝜋 2𝜋 24𝜋 12 6
12

360° ? 360 11𝜋 11∗180 11∗90 11∗45 11∗15 165
c. = 11 → ∗ 24 =? = = = = = = 82,5°
2𝜋 𝜋 2𝜋 24 12 6 2 2
24

360° ? 360 15𝜋 15∗360𝜋 15∗180 15∗90 15∗45 675
d. 2𝜋
= 15 → 2𝜋
∗ 8 =? = 16𝜋
= 8
= 4
= 2
= 2
= 337,5°
8
𝜋

, 360° ? 360 23𝜋 23∗360𝜋 23∗180 23∗90 23∗45
e. = 23 → ∗ 12 =? = = = = = 23 ∗ 15 = 345°
2𝜋 𝜋 2𝜋 24𝜋 12 6 3
12




17.6

360° ? 360 71𝜋 71∗360𝜋 71∗180 71∗30 71∗5 355
a. = 71 → ∗ 72 =? = = = = = = 177,5°
2𝜋 𝜋 2𝜋 144𝜋 72 12 2 2
72

360° ? 360 41𝜋 41∗180 41∗90 41∗30 41∗15 615
b. = 41 → ∗ =? = = = = = = 307,5°
2𝜋 𝜋 2𝜋 24 24 12 4 2 2
24

360° ? 360 25𝜋 25∗180 25∗10
c. = 25 → ∗ 18 =? = = = 250°
2𝜋 𝜋 2𝜋 18 1
18

360° ? 360 13𝜋 13∗180 13∗90 13∗30 13∗15 195
d. 2𝜋
= 13 → 2𝜋
∗ 24 =? = 24
= 12
= 4
= 2
= 2
= 97,5°
𝜋
24

360° ? 360 31𝜋 31∗180 31∗30
e. = 31 → ∗ 36 =? = = = 31 ∗ 5 = 155°
2𝜋 𝜋 2𝜋 36 6
36




17.7

CW = clockwise

CCW = counter clockwise

a.

−30° 𝑘𝑜𝑚𝑡 𝑜𝑣𝑒𝑟𝑒𝑒𝑛 𝑚𝑒𝑡 𝑒𝑒𝑛 𝑑𝑟𝑎𝑎𝑖𝑖𝑛𝑔 𝑟𝑒𝑐ℎ𝑡𝑠𝑜𝑚 (𝑚𝑒𝑡 𝑑𝑒 𝑘𝑙𝑜𝑘 𝑚𝑒𝑒, 𝑣𝑎𝑛𝑎𝑓 𝑛𝑢 𝐶𝑊 (𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒))

𝐷𝑖𝑡 𝑘𝑜𝑚𝑡 𝑤𝑒𝑒𝑟 𝑜𝑣𝑒𝑟𝑒𝑒𝑛 𝑚𝑒𝑡 𝑒𝑒𝑛 𝑑𝑟𝑎𝑎𝑖𝑖𝑛𝑔 𝑣𝑎𝑛 360° − 30° (𝑡𝑒𝑘𝑒𝑛 𝑑𝑖𝑡 𝑜𝑝 𝑑𝑒 𝑐𝑖𝑟𝑘𝑒𝑙)

360 − 30 = 330° CCW (counter clockwise)

b. 445° 𝑘𝑜𝑚𝑡 𝑛𝑒𝑒𝑟 𝑜𝑝 𝑚𝑒𝑒𝑟 𝑑𝑎𝑛 éé𝑛 𝑟𝑜𝑛𝑑𝑗𝑒 𝑜𝑚 𝑑𝑒 𝑒𝑒𝑛ℎ𝑒𝑖𝑑𝑠𝑐𝑖𝑟𝑘𝑒𝑙: 360°

𝐻𝑎𝑎𝑙 ℎ𝑒𝑡 𝑒𝑒𝑟𝑠𝑡𝑒 𝑟𝑜𝑛𝑑𝑗𝑒 𝑒𝑟𝑣𝑎𝑛 𝑎𝑓: 445 − 360 = 85° 𝑖𝑠 𝑑𝑎𝑛 𝑤𝑎𝑡 𝑒𝑟 𝑜𝑣𝑒𝑟𝑏𝑙𝑖𝑗𝑓𝑡

𝐷𝑖𝑡 𝑘𝑜𝑚𝑡 𝑛𝑒𝑒𝑟 𝑜𝑝 𝑝𝑟𝑒𝑐𝑖𝑒𝑠 𝑑𝑒𝑧𝑒𝑙𝑓𝑑𝑒 ℎ𝑜𝑒𝑘, 𝑤𝑎𝑛𝑡 𝑛𝑎 𝑑𝑒 𝑒𝑒𝑟𝑠𝑡𝑒 360° 𝑏𝑒𝑔𝑖𝑛𝑡 𝑑𝑒 𝑑𝑟𝑎𝑎𝑖𝑖𝑛𝑔 𝑜𝑝𝑛𝑖𝑒𝑢𝑤

c. −160° 𝑤𝑜𝑟𝑑𝑡 𝑜𝑝 𝑑𝑒𝑧𝑒𝑙𝑓𝑑𝑒 𝑚𝑎𝑛𝑖𝑒𝑟 𝑏𝑒𝑝𝑎𝑎𝑙𝑑 𝑎𝑙𝑠 𝑏𝑖𝑗 𝑎.

360 − 160 = 200°

𝑇𝑒𝑘𝑒𝑛 𝑜𝑜𝑘 𝑑𝑖𝑡 𝑜𝑝 𝑑𝑒 𝑒𝑒𝑛ℎ𝑒𝑖𝑑𝑠𝑐𝑖𝑟𝑘𝑒𝑙 𝑜𝑚 𝑒𝑒𝑛 𝑏𝑒𝑒𝑙𝑑 𝑡𝑒 𝑣𝑜𝑟𝑚𝑒𝑛.

d. 700° 𝑖𝑠 𝑜𝑜𝑘 𝑤𝑒𝑒𝑟 𝑚𝑒𝑒𝑟 𝑑𝑎𝑛 éé𝑛 𝑟𝑜𝑛𝑑𝑗𝑒: ℎ𝑎𝑎𝑙 𝑑𝑎𝑎𝑟𝑜𝑚 ℎ𝑒𝑡 𝑒𝑒𝑟𝑠𝑡𝑒 𝑟𝑜𝑛𝑑𝑗𝑒 𝑒𝑟𝑎𝑓.

700 − 360 = 340°

e. 515° − 360° = 155°